Unipotent cells in Kac-Moody groups Bernard Leclerc joint with - - PowerPoint PPT Presentation

Unipotent cells in Kac-Moody groups

Bernard Leclerc joint with Christof Geiss and Jan Schr¨

• er

ICTP Trieste, February 2010

Aims of our joint project :

Aims of our joint project : (1) Describe (following Berenstein-Fomin-Zelevinsky) cluster algebra structures on coordinate rings of unipotent cells of Kac-Moody groups.

Aims of our joint project : (1) Describe (following Berenstein-Fomin-Zelevinsky) cluster algebra structures on coordinate rings of unipotent cells of Kac-Moody groups. (2) Categorify these coordinate rings using preprojective algebras.

Aims of our joint project : (1) Describe (following Berenstein-Fomin-Zelevinsky) cluster algebra structures on coordinate rings of unipotent cells of Kac-Moody groups. (2) Categorify these coordinate rings using preprojective algebras. (3) Understand better this class of cluster algebras by means of semicanonical bases.

Aims of our joint project : (1) Describe (following Berenstein-Fomin-Zelevinsky) cluster algebra structures on coordinate rings of unipotent cells of Kac-Moody groups. (2) Categorify these coordinate rings using preprojective algebras. (3) Understand better this class of cluster algebras by means of semicanonical bases. In this talk : Lie-theoretic preliminaries for lectures of Schr¨

• er and Geiss.

Plan

Plan

Kac-Moody algebras

Plan

Kac-Moody algebras Unipotent subgroups and Kac-Moody groups

Plan

Kac-Moody algebras Unipotent subgroups and Kac-Moody groups Unipotent cells and their coordinate rings

Plan

Kac-Moody algebras Unipotent subgroups and Kac-Moody groups Unipotent cells and their coordinate rings Preprojective algebras and semicanonical bases

Kac-Moody algebras

Cartan matrix

Cartan matrix

Q : acyclic quiver with vertex set I = {1, 2, . . . , n}.

Cartan matrix

Q : acyclic quiver with vertex set I = {1, 2, . . . , n}. Γ : underlying unoriented graph.

Cartan matrix

Q : acyclic quiver with vertex set I = {1, 2, . . . , n}. Γ : underlying unoriented graph. C = [cij] where cij = 2δij − ♯{edges between i and j in Γ}.

Cartan matrix

Q : acyclic quiver with vertex set I = {1, 2, . . . , n}. Γ : underlying unoriented graph. C = [cij] where cij = 2δij − ♯{edges between i and j in Γ}. Example: Q = 1

2 3

Γ = 1 2 3 C =   2 −2 −2 2 −1 −1 2  

Kac-Moody algebra

Kac-Moody algebra

h : C-vector space of dimension 2n − rk C

Kac-Moody algebra

h : C-vector space of dimension 2n − rk C {h1, . . . , hn} ⊂ h, {α1, . . . , αn} ⊂ h∗ linearly independent s.t. αi(hj) = cij

Kac-Moody algebra

h : C-vector space of dimension 2n − rk C {h1, . . . , hn} ⊂ h, {α1, . . . , αn} ⊂ h∗ linearly independent s.t. αi(hj) = cij g : Lie algebra over C with generators ei, fi (i ∈ I), h ∈ h, and relations : [h, h′] = 0, [h, ei] = αi(h)ei, [h,fi] = −αi(h)fi, [ei, fj] = δijhi,

ad(ei)1−cij(ej) = ad(fi)1−cij(fj) = 0, (i = j).

Kac-Moody algebra

h : C-vector space of dimension 2n − rk C {h1, . . . , hn} ⊂ h, {α1, . . . , αn} ⊂ h∗ linearly independent s.t. αi(hj) = cij g : Lie algebra over C with generators ei, fi (i ∈ I), h ∈ h, and relations : [h, h′] = 0, [h, ei] = αi(h)ei, [h,fi] = −αi(h)fi, [ei, fj] = δijhi,

ad(ei)1−cij(ej) = ad(fi)1−cij(fj) = 0, (i = j).

n+ = n := ei | i ∈ I, n− := fi | i ∈ I.

Weyl group and Roots

Weyl group and Roots

W : subgroup of GL(h∗) generated by the reflexions si (i ∈ I): si(α) = α − α(hi)αi.

Weyl group and Roots

W : subgroup of GL(h∗) generated by the reflexions si (i ∈ I): si(α) = α − α(hi)αi. W is a Coxeter group with length function w → ℓ(w).

Weyl group and Roots

W : subgroup of GL(h∗) generated by the reflexions si (i ∈ I): si(α) = α − α(hi)αi. W is a Coxeter group with length function w → ℓ(w). For α ∈ h∗, let gα := {x ∈ g | [h, x] = α(h)x, h ∈ h}.

Weyl group and Roots

W : subgroup of GL(h∗) generated by the reflexions si (i ∈ I): si(α) = α − α(hi)αi. W is a Coxeter group with length function w → ℓ(w). For α ∈ h∗, let gα := {x ∈ g | [h, x] = α(h)x, h ∈ h}. ∆ := {α ∈ h∗ | α = 0 and gα = 0} : root system of g.

Weyl group and Roots

W : subgroup of GL(h∗) generated by the reflexions si (i ∈ I): si(α) = α − α(hi)αi. W is a Coxeter group with length function w → ℓ(w). For α ∈ h∗, let gα := {x ∈ g | [h, x] = α(h)x, h ∈ h}. ∆ := {α ∈ h∗ | α = 0 and gα = 0} : root system of g. Set R+ =

i∈I Nαi, and ∆+ := ∆ ∩ R+. We have

∆ = ∆+ ⊔ (−∆+).

Weyl group and Roots

W : subgroup of GL(h∗) generated by the reflexions si (i ∈ I): si(α) = α − α(hi)αi. W is a Coxeter group with length function w → ℓ(w). For α ∈ h∗, let gα := {x ∈ g | [h, x] = α(h)x, h ∈ h}. ∆ := {α ∈ h∗ | α = 0 and gα = 0} : root system of g. Set R+ =

i∈I Nαi, and ∆+ := ∆ ∩ R+. We have

∆ = ∆+ ⊔ (−∆+). W acts on ∆. Define ∆re := W {α1, . . . , αn}.

Weyl group and Roots

W : subgroup of GL(h∗) generated by the reflexions si (i ∈ I): si(α) = α − α(hi)αi. W is a Coxeter group with length function w → ℓ(w). For α ∈ h∗, let gα := {x ∈ g | [h, x] = α(h)x, h ∈ h}. ∆ := {α ∈ h∗ | α = 0 and gα = 0} : root system of g. Set R+ =

i∈I Nαi, and ∆+ := ∆ ∩ R+. We have

∆ = ∆+ ⊔ (−∆+). W acts on ∆. Define ∆re := W {α1, . . . , αn}. For w ∈ W , let ∆w := {α ∈ ∆+ | w(α) ∈ ∆−} ⊂ ∆re.

Weyl group and Roots

W : subgroup of GL(h∗) generated by the reflexions si (i ∈ I): si(α) = α − α(hi)αi. W is a Coxeter group with length function w → ℓ(w). For α ∈ h∗, let gα := {x ∈ g | [h, x] = α(h)x, h ∈ h}. ∆ := {α ∈ h∗ | α = 0 and gα = 0} : root system of g. Set R+ =

i∈I Nαi, and ∆+ := ∆ ∩ R+. We have

∆ = ∆+ ⊔ (−∆+). W acts on ∆. Define ∆re := W {α1, . . . , αn}. For w ∈ W , let ∆w := {α ∈ ∆+ | w(α) ∈ ∆−} ⊂ ∆re. Set n(w) :=

• α∈∆w

gα ⊂ n.

Weyl group and Roots

W : subgroup of GL(h∗) generated by the reflexions si (i ∈ I): si(α) = α − α(hi)αi. W is a Coxeter group with length function w → ℓ(w). For α ∈ h∗, let gα := {x ∈ g | [h, x] = α(h)x, h ∈ h}. ∆ := {α ∈ h∗ | α = 0 and gα = 0} : root system of g. Set R+ =

i∈I Nαi, and ∆+ := ∆ ∩ R+. We have

∆ = ∆+ ⊔ (−∆+). W acts on ∆. Define ∆re := W {α1, . . . , αn}. For w ∈ W , let ∆w := {α ∈ ∆+ | w(α) ∈ ∆−} ⊂ ∆re. Set n(w) :=

• α∈∆w

gα ⊂ n. We have ♯∆w = dim n(w) = ℓ(w).

An example

An example

Q = 1

2

An example

Q = 1

2

W = s1, s2 | s2

1 = s2 2 = 1.

An example

Q = 1

2

W = s1, s2 | s2

1 = s2 2 = 1.

w = s2s1s2s1.

An example

Q = 1

2

W = s1, s2 | s2

1 = s2 2 = 1.

w = s2s1s2s1. ∆w := {α1, 2α1 + α2, 3α1 + 2α2, 4α1 + 3α2}.

An example

Q = 1

2

W = s1, s2 | s2

1 = s2 2 = 1.

w = s2s1s2s1. ∆w := {α1, 2α1 + α2, 3α1 + 2α2, 4α1 + 3α2}. n(w) := Spane1, [e1, [e2, e1]], [e1, [e2, [e1, [e2, e1]]]], [e1, [e2, [e1, [e2, [e1, [e2, e1]]]]]].

Unipotent groups and Kac-Moody groups

The group N

The group N

U(n) : universal enveloping algebra of n. It is generated by ei (i ∈ I), with relations

1−cij

• k=0

(−1)ke(k)

i

eje(1−cij−k)

i

= 0, (i = j).

The group N

U(n) : universal enveloping algebra of n. It is generated by ei (i ∈ I), with relations

1−cij

• k=0

(−1)ke(k)

i

eje(1−cij−k)

i

= 0, (i = j). U(n) is a cocommutative Hopf algebra, with an R+-grading given by deg(ei) := αi.

The group N

U(n) : universal enveloping algebra of n. It is generated by ei (i ∈ I), with relations

1−cij

• k=0

(−1)ke(k)

i

eje(1−cij−k)

i

= 0, (i = j). U(n) is a cocommutative Hopf algebra, with an R+-grading given by deg(ei) := αi. Let U(n)∗

gr := d∈R+ U(n)∗ d be the graded dual. This is a

commutative Hopf algebra.

The group N

U(n) : universal enveloping algebra of n. It is generated by ei (i ∈ I), with relations

1−cij

• k=0

(−1)ke(k)

i

eje(1−cij−k)

i

= 0, (i = j). U(n) is a cocommutative Hopf algebra, with an R+-grading given by deg(ei) := αi. Let U(n)∗

gr := d∈R+ U(n)∗ d be the graded dual. This is a

commutative Hopf algebra. Let N := maxSpec(U(n)∗

gr) = Hom alg(U(n)∗ gr, C). This is the

pro-unipotent pro-group with Lie algebra

• n =
• α∈∆+

gα.

The group N

U(n) : universal enveloping algebra of n. It is generated by ei (i ∈ I), with relations

1−cij

• k=0

(−1)ke(k)

i

eje(1−cij−k)

i

= 0, (i = j). U(n) is a cocommutative Hopf algebra, with an R+-grading given by deg(ei) := αi. Let U(n)∗

gr := d∈R+ U(n)∗ d be the graded dual. This is a

commutative Hopf algebra. Let N := maxSpec(U(n)∗

gr) = Hom alg(U(n)∗ gr, C). This is the

pro-unipotent pro-group with Lie algebra

• n =
• α∈∆+

gα. By construction, U(n)∗

gr = C[N].

The unipotent subgroup N(w)

The unipotent subgroup N(w)

Let N(w) be the subgroup of N with Lie algebra n(w).

The unipotent subgroup N(w)

Let N(w) be the subgroup of N with Lie algebra n(w). Let N′(w) be the subgroup of N with Lie algebra n′(w) :=

• α∈∆w

gα ⊂ n.

The unipotent subgroup N(w)

Let N(w) be the subgroup of N with Lie algebra n(w). Let N′(w) be the subgroup of N with Lie algebra n′(w) :=

• α∈∆w

gα ⊂ n. Multiplication yields a bijection N(w) × N′(w) ∼ → N.

The unipotent subgroup N(w)

Let N(w) be the subgroup of N with Lie algebra n(w). Let N′(w) be the subgroup of N with Lie algebra n′(w) :=

• α∈∆w

gα ⊂ n. Multiplication yields a bijection N(w) × N′(w) ∼ → N. Proposition The coordinate ring C[N(w)] is isomorphic to the invariant subring C[N]N′(w) =

• f ∈ C[N] | f (nn′) = f (n), n ∈ N, n′ ∈ N(w)

The Kac-Moody group G

The Kac-Moody group G

Let G be the group attached to g by Kac-Peterson.

The Kac-Moody group G

Let G be the group attached to g by Kac-Peterson. This is an affine ind-variety. It has a refined Tits system (G, NormG(H), N+, N−, H), where Lie(H) = h, Lie(N+) = n+, and Lie(N−) = n−.

The Kac-Moody group G

Let G be the group attached to g by Kac-Peterson. This is an affine ind-variety. It has a refined Tits system (G, NormG(H), N+, N−, H), where Lie(H) = h, Lie(N+) = n+, and Lie(N−) = n−. Note : In general N ⊂ G. They can both be regarded as subgroups of a bigger group G max constructed by Tits. Then N+ = N ∩ G.

The Kac-Moody group G

Let G be the group attached to g by Kac-Peterson. This is an affine ind-variety. It has a refined Tits system (G, NormG(H), N+, N−, H), where Lie(H) = h, Lie(N+) = n+, and Lie(N−) = n−. Note : In general N ⊂ G. They can both be regarded as subgroups of a bigger group G max constructed by Tits. Then N+ = N ∩ G. We have NormG(H)/H ∼ = W .

The Kac-Moody group G

Let G be the group attached to g by Kac-Peterson. This is an affine ind-variety. It has a refined Tits system (G, NormG(H), N+, N−, H), where Lie(H) = h, Lie(N+) = n+, and Lie(N−) = n−. Note : In general N ⊂ G. They can both be regarded as subgroups of a bigger group G max constructed by Tits. Then N+ = N ∩ G. We have NormG(H)/H ∼ = W . For i ∈ I, put si := exp(fi) exp(−ei) exp(fi) ∈ NormG(H).

The Kac-Moody group G

Let G be the group attached to g by Kac-Peterson. This is an affine ind-variety. It has a refined Tits system (G, NormG(H), N+, N−, H), where Lie(H) = h, Lie(N+) = n+, and Lie(N−) = n−. Note : In general N ⊂ G. They can both be regarded as subgroups of a bigger group G max constructed by Tits. Then N+ = N ∩ G. We have NormG(H)/H ∼ = W . For i ∈ I, put si := exp(fi) exp(−ei) exp(fi) ∈ NormG(H). For w = sir · · · si1 with ℓ(w) = r, put w = sir · · · si1, a representative of w in NormG(H).

Examples

Examples

If Q is of Dynkin type Xn, then G = G Xn(C) is a connected simply-connected algebraic group of type Xn over C. (Ex: If Xn = An, G = SL(n + 1, C).)

Examples

If Q is of Dynkin type Xn, then G = G Xn(C) is a connected simply-connected algebraic group of type Xn over C. (Ex: If Xn = An, G = SL(n + 1, C).) If Q is of affine Dynkin type Xn, then G is a central extension by C∗ of G Xn(C[z, z−1]).

Examples

If Q is of Dynkin type Xn, then G = G Xn(C) is a connected simply-connected algebraic group of type Xn over C. (Ex: If Xn = An, G = SL(n + 1, C).) If Q is of affine Dynkin type Xn, then G is a central extension by C∗ of G Xn(C[z, z−1]). Moreover, N+ ≃ {g ∈ G Xn(C[z]) | g|z=0 ∈ NXn(C)}.

Examples

If Q is of Dynkin type Xn, then G = G Xn(C) is a connected simply-connected algebraic group of type Xn over C. (Ex: If Xn = An, G = SL(n + 1, C).) If Q is of affine Dynkin type Xn, then G is a central extension by C∗ of G Xn(C[z, z−1]). Moreover, N+ ≃ {g ∈ G Xn(C[z]) | g|z=0 ∈ NXn(C)}. If Q is wild, no “concrete” realization of G is known.

Unipotent cells

Generalized minors

Generalized minors

Let G0 = N−HN+.

Generalized minors

Let G0 = N−HN+. Proposition G0 is open dense in G. Every g ∈ G0 has a unique factorization g = [g]−[g]0[g]+ with [g]− ∈ N−, [g]0 ∈ H, [g]+ ∈ N+.

Generalized minors

Let G0 = N−HN+. Proposition G0 is open dense in G. Every g ∈ G0 has a unique factorization g = [g]−[g]0[g]+ with [g]− ∈ N−, [g]0 ∈ H, [g]+ ∈ N+. For i ∈ I, let ̟i ∈ h∗ s. t. ̟i(hj) = δij.

Generalized minors

Let G0 = N−HN+. Proposition G0 is open dense in G. Every g ∈ G0 has a unique factorization g = [g]−[g]0[g]+ with [g]− ∈ N−, [g]0 ∈ H, [g]+ ∈ N+. For i ∈ I, let ̟i ∈ h∗ s. t. ̟i(hj) = δij. Let x → x̟i denote the corresponding character of H.

Generalized minors

Let G0 = N−HN+. Proposition G0 is open dense in G. Every g ∈ G0 has a unique factorization g = [g]−[g]0[g]+ with [g]− ∈ N−, [g]0 ∈ H, [g]+ ∈ N+. For i ∈ I, let ̟i ∈ h∗ s. t. ̟i(hj) = δij. Let x → x̟i denote the corresponding character of H. There is a unique regular function ∆̟i,̟i on G such that ∆̟i,̟i(g) = [g]̟i (g ∈ G0).

Generalized minors

Let G0 = N−HN+. Proposition G0 is open dense in G. Every g ∈ G0 has a unique factorization g = [g]−[g]0[g]+ with [g]− ∈ N−, [g]0 ∈ H, [g]+ ∈ N+. For i ∈ I, let ̟i ∈ h∗ s. t. ̟i(hj) = δij. Let x → x̟i denote the corresponding character of H. There is a unique regular function ∆̟i,̟i on G such that ∆̟i,̟i(g) = [g]̟i (g ∈ G0). For w ∈ W , set ∆̟i,w(̟i)(g) := ∆̟i,̟i(gw).

Generalized minors

Let G0 = N−HN+. Proposition G0 is open dense in G. Every g ∈ G0 has a unique factorization g = [g]−[g]0[g]+ with [g]− ∈ N−, [g]0 ∈ H, [g]+ ∈ N+. For i ∈ I, let ̟i ∈ h∗ s. t. ̟i(hj) = δij. Let x → x̟i denote the corresponding character of H. There is a unique regular function ∆̟i,̟i on G such that ∆̟i,̟i(g) = [g]̟i (g ∈ G0). For w ∈ W , set ∆̟i,w(̟i)(g) := ∆̟i,̟i(gw). G0 = {g ∈ G | ∆̟i,̟i(g) = 0, i ∈ I}.

The unipotent cell Nw

The unipotent cell Nw

Let B− = N−H.

The unipotent cell Nw

Let B− = N−H. The group G has a Bruhat decomposition G =

• w∈W

B−wB−.

The unipotent cell Nw

Let B− = N−H. The group G has a Bruhat decomposition G =

• w∈W

B−wB−. For w ∈ W , define Nw := N+ B−wB−.

The unipotent cell Nw

Let B− = N−H. The group G has a Bruhat decomposition G =

• w∈W

B−wB−. For w ∈ W , define Nw := N+ B−wB−. Let Ow := {n ∈ N(w) | ∆̟i,w−1(̟i)(n) = 0, i ∈ I}.

The unipotent cell Nw

Let B− = N−H. The group G has a Bruhat decomposition G =

• w∈W

B−wB−. For w ∈ W , define Nw := N+ B−wB−. Let Ow := {n ∈ N(w) | ∆̟i,w−1(̟i)(n) = 0, i ∈ I}. Proposition We have an isomorphism Ow

∼

→ Nw. It follows that C[Nw] is the localization of C[N(w)] ≃ C[N]N′(w) at ∆w :=

• i∈I

∆̟i,w−1(̟i).

Concrete calculations

Concrete calculations

Set xi(t) := exp(tei) (t ∈ C, i ∈ I).

Concrete calculations

Set xi(t) := exp(tei) (t ∈ C, i ∈ I). Let w = si1 · · · sir be a reduced decomposition.

Concrete calculations

Set xi(t) := exp(tei) (t ∈ C, i ∈ I). Let w = si1 · · · sir be a reduced decomposition. The image of the map (C∗)r → N given by (t1, . . . , tr) → xi1(t1) · · · xir (tr) is a dense subset of Nw.

Concrete calculations

Set xi(t) := exp(tei) (t ∈ C, i ∈ I). Let w = si1 · · · sir be a reduced decomposition. The image of the map (C∗)r → N given by (t1, . . . , tr) → xi1(t1) · · · xir (tr) is a dense subset of Nw. If f ∈ C[Nw], then (t1, . . . , tr) → f (xi1(t1) · · · xir (tr)) is a polynomial function, which completely determines f .

Concrete calculations

Set xi(t) := exp(tei) (t ∈ C, i ∈ I). Let w = si1 · · · sir be a reduced decomposition. The image of the map (C∗)r → N given by (t1, . . . , tr) → xi1(t1) · · · xir (tr) is a dense subset of Nw. If f ∈ C[Nw], then (t1, . . . , tr) → f (xi1(t1) · · · xir (tr)) is a polynomial function, which completely determines f . Problem How to calculate f (xi1(t1) · · · xir (tr)), for example when f is a generalized minor ?

Preprojective algebras and semicanonical bases

The preprojective algebra

The preprojective algebra

Λ : the preprojective algebra attached to Q.

The preprojective algebra

Λ : the preprojective algebra attached to Q. nil(Λ) : category of finite-dimensional nilpotent Λ-modules.

The preprojective algebra

Λ : the preprojective algebra attached to Q. nil(Λ) : category of finite-dimensional nilpotent Λ-modules. For M ∈ nil(Λ) and i := (i1, . . . , ik), let FM,i be the variety of type i composition series of M.

The preprojective algebra

Λ : the preprojective algebra attached to Q. nil(Λ) : category of finite-dimensional nilpotent Λ-modules. For M ∈ nil(Λ) and i := (i1, . . . , ik), let FM,i be the variety of type i composition series of M. χM,i := χ(FM,i) ∈ Z (Euler characteristic).

The preprojective algebra

Λ : the preprojective algebra attached to Q. nil(Λ) : category of finite-dimensional nilpotent Λ-modules. For M ∈ nil(Λ) and i := (i1, . . . , ik), let FM,i be the variety of type i composition series of M. χM,i := χ(FM,i) ∈ Z (Euler characteristic). Theorem (Lusztig, Geiss-L-Schr¨

• er)

There exits a unique ϕM ∈ C[N] such that for all j = (j1, . . . , jk) ϕM(xj1(t1) · · · xjk(tk)) =

• a∈Nk

χM,ja t1a1 · · · tkak a1! · · · ak! where ja = (j1, . . . , j1

• a1

, . . . , jk, . . . , jk

• ak

)

An example

An example

Q = 1

2

An example

Q = 1

2

M1 = 1

• 1
• 2

M2 = 2

• 2
• 2
• 1
• 1
• 2

An example

Q = 1

2

M1 = 1

• 1
• 2

M2 = 2

• 2
• 2
• 1
• 1
• 2

ϕM1(x2(t1)x1(t2)x2(t3)x1(t4)) = t1t2

2 + 2t1t2t4 + t1t2 4 + t3t2 4

An example

Q = 1

2

M1 = 1

• 1
• 2

M2 = 2

• 2
• 2
• 1
• 1
• 2

ϕM1(x2(t1)x1(t2)x2(t3)x1(t4)) = t1t2

2 + 2t1t2t4 + t1t2 4 + t3t2 4

ϕM2(x2(t1)x1(t2)x2(t3)x1(t4)) = t1t2

2t3 3

Generalized minors

Generalized minors

For M ∈ Mod Λ and i ∈ I, let soc i(M) be the Si-isotypic component of soc M.

Generalized minors

For M ∈ Mod Λ and i ∈ I, let soc i(M) be the Si-isotypic component of soc M. For i = (i1, . . . , im) there is a unique sequence 0 = M0 ⊆ M1 ⊆ · · · ⊆ Mm ⊆ M such that soc ik(M/Mk−1) ∼ = Mk/Mk−1 for 1 ≤ k ≤ m. Define soc i(M) = Mm.

Generalized minors

For M ∈ Mod Λ and i ∈ I, let soc i(M) be the Si-isotypic component of soc M. For i = (i1, . . . , im) there is a unique sequence 0 = M0 ⊆ M1 ⊆ · · · ⊆ Mm ⊆ M such that soc ik(M/Mk−1) ∼ = Mk/Mk−1 for 1 ≤ k ≤ m. Define soc i(M) = Mm.

• Ik : injective envelope of Sk (infinite-dimensional).

Generalized minors

For M ∈ Mod Λ and i ∈ I, let soc i(M) be the Si-isotypic component of soc M. For i = (i1, . . . , im) there is a unique sequence 0 = M0 ⊆ M1 ⊆ · · · ⊆ Mm ⊆ M such that soc ik(M/Mk−1) ∼ = Mk/Mk−1 for 1 ≤ k ≤ m. Define soc i(M) = Mm.

• Ik : injective envelope of Sk (infinite-dimensional).

Proposition (Geiss-L-Schr¨

• er)

Let i be a reduced word for w−1 ∈ W . Then ∆̟k,w(̟k) = ϕsoc i(b

Ik).

The semicanonical basis

The semicanonical basis

d ∈ NI : dimension vector for Λ.

The semicanonical basis

d ∈ NI : dimension vector for Λ. Λd : variety of nilpotent Λ-modules of dimension vector d.

The semicanonical basis

d ∈ NI : dimension vector for Λ. Λd : variety of nilpotent Λ-modules of dimension vector d. Irr(Λd) : set of irreducible components of Λd.

The semicanonical basis

d ∈ NI : dimension vector for Λ. Λd : variety of nilpotent Λ-modules of dimension vector d. Irr(Λd) : set of irreducible components of Λd. Irr(Λ) :=

d∈NI Irr(Λd).

The semicanonical basis

d ∈ NI : dimension vector for Λ. Λd : variety of nilpotent Λ-modules of dimension vector d. Irr(Λd) : set of irreducible components of Λd. Irr(Λ) :=

d∈NI Irr(Λd).

For Z ∈ Irr(Λ), there is a dense U ⊂ Z such that M → ϕM is constant on U. Set ϕZ := ϕM for M ∈ U.

The semicanonical basis

d ∈ NI : dimension vector for Λ. Λd : variety of nilpotent Λ-modules of dimension vector d. Irr(Λd) : set of irreducible components of Λd. Irr(Λ) :=

d∈NI Irr(Λd).

For Z ∈ Irr(Λ), there is a dense U ⊂ Z such that M → ϕM is constant on U. Set ϕZ := ϕM for M ∈ U. Theorem (Lusztig, Geiss-L-Schr¨

• er)

S∗ := {ϕZ | Z ∈ Irr(Λ)} is a basis of C[N].

The semicanonical basis

d ∈ NI : dimension vector for Λ. Λd : variety of nilpotent Λ-modules of dimension vector d. Irr(Λd) : set of irreducible components of Λd. Irr(Λ) :=

d∈NI Irr(Λd).

For Z ∈ Irr(Λ), there is a dense U ⊂ Z such that M → ϕM is constant on U. Set ϕZ := ϕM for M ∈ U. Theorem (Lusztig, Geiss-L-Schr¨

• er)

S∗ := {ϕZ | Z ∈ Irr(Λ)} is a basis of C[N]. S∗ is dual to Lusztig’s semicanonical basis of U(n).

The semicanonical basis

d ∈ NI : dimension vector for Λ. Λd : variety of nilpotent Λ-modules of dimension vector d. Irr(Λd) : set of irreducible components of Λd. Irr(Λ) :=

d∈NI Irr(Λd).

For Z ∈ Irr(Λ), there is a dense U ⊂ Z such that M → ϕM is constant on U. Set ϕZ := ϕM for M ∈ U. Theorem (Lusztig, Geiss-L-Schr¨

• er)

S∗ := {ϕZ | Z ∈ Irr(Λ)} is a basis of C[N]. S∗ is dual to Lusztig’s semicanonical basis of U(n). The generalized minors ∆̟k,w(̟k) belong to S∗.

Aims

Aims

Categorify C[N(w)] and C[Nw], and obtain cluster algebra structures.

Aims

Categorify C[N(w)] and C[Nw], and obtain cluster algebra structures. Study these cluster algebras by means of preprojective algebras and semicanonical bases.

Aims

Categorify C[N(w)] and C[Nw], and obtain cluster algebra structures. Study these cluster algebras by means of preprojective algebras and semicanonical bases. → talks of Jan Schr¨

• er and Christof Geiss.